A central object in the study of finite-type surfaces and their mapping class groups is the curve complex \(S)\) , which was introduced by Harvey [1]}. The vertices of \(S)\)  represent curves on \(S\)  and sets of vertices that have disjoint representatives bound simplices. The curve complex is flag and it is also referred to as the curve graph. Many variants of \(S)\)  have also been considered—for example, by replacing curves with subsurfaces or by considering only separating or nonseparating curves. When \(S\)  is of finite type, \(S)\)  has infinite diameter when equipped with the path metric, and its geometry is Gromov hyperbolic, as shown by Masur–Minsky [2]}. Further, the orbits of the natural action of \(\operatorname{Map}(S)\)  on \(S)\)  have infinite diameter, and analyzing these orbits gives information about the Nielsen–Thurston type of a mapping class.
